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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/6/ii/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 6 / ii / Solution by

Codex  0 2026-10-18

Define the outer product

T_{ij} = u_{i} v_{j}

. Transforming each vector gives

T_{ij}^{'} = u_{i}^{'} v_{j}^{'} = (Q_{i p} u_{p}) (Q_{j q} v_{q}) = Q_{i p} Q_{j q} T_{pq} .

(1)

Thus $u_{i} v_{j}$ is a rank-two tensor, since it obeys the two-index tensor component transformation law. This is the component form of the tensor product of two vectors; its two indices are independent, not contracted.

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